Page 28 - Azerbaijan State University of Economics

Salah Salimian, Kiumars Shahbazi, Mahdi M. Beknazar: Locating The Sale Agents In Spoke Model
Through Uniform Distribution Of Consumers

D

M

M

π = ̂ ( − p ) − F = γ ⇒ ( − ) ( − p ) − F = γ

1
Where there are s streets on which the consumers are uniformly distributed ( );

therefore, the following equation will be achieved:
2

M
⇒ p = ( − ( ) − F − γ) (2)

( − )

The monopolist profit in this state equals to:

2
π = ( − ) p = − ( ) − F − γ (3)

M
M

Now, we assume that the sale agent is located on one of these streets, in this state, n
consumer located on the same street where the sale agent is located (in case of
1
D
purchase) will have a surplus of p + − B (purchase from the sale agent on its
2
1
street), other consumers (in case of purchase) will also have a surplus of B − − p
D
2
(these consumers are inevitable to pass city center and refer to another street where
the sale agent is located due to lack of sale agent on their own street or center). The
maximum price that the monopolist could demand from sale agent will be achieved
from the following equation:

1 ( − 1)

D
π = [ ̂ ( − ) + ̂ ( − )] − =

1 1 ( −1) 1

⇒ [ ( + − ) ( − ) + ( − − ) ( − )] − =
2 2
(4)

Therefore:
2
1

2 + + − (2 − )( ) − + s − s B
M
p = 1 2 (5)

( − 2) − 1 + 2 − +
2

Therefore, the monopolist profit in this state equals to:

1

M
π = (2 − )( ) − 2 + − s + s B − − (6)

2

2

By comparison of equations 3 and 6, the monopolist will launch sale agent at city
center rather than margin when the following relation is established:

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